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A compactness result for a second-order variational discrete model

Published online by Cambridge University Press:  23 November 2011

Andrea Braides ,
Anneliese Defranceschi  and
Enrico Vitali
Andrea Braides
Affiliation:
Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, via della Ricerca Scientifica, 00133 Rome, Italy. braides@mat.uniroma2.it
Anneliese Defranceschi
Affiliation:
Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38123 Povo, Italy
Enrico Vitali
Affiliation:
Dipartimento di Matematica ‘F. Casorati’, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy
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Abstract

We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions with bounded hessian, and we give an upper and a lower bound in terms of the Blake and Zisserman energy. We prove a sharp bound by exhibiting the discrete-to-continuous Γ-limit for a special class of functions, showing the appearance new ‘shear’ terms in the energy, which are a genuinely two-dimensional effect.

Keywords

Computer visionfinite-difference schemesgamma-convergencefree-discontinuity problems
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 2 , March 2012 , pp. 389 - 410
Copyright
© EDP Sciences, SMAI, 2011

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